Algebraic and Geometric Ideas in the Theory of Discrete Optimization
Algebraic and Geometric Ideas in the Theory of Discrete Optimization
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离散优化理论中的代数和几何思想
DOI：
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发表时间：
2012
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通讯作者：
M. Köppe
中科院分区：
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作者：
J. D. Loera;R. Hemmecke;M. Köppe
本书介绍了离散优化的数学理论的最新进展，尤其是由代数几何学，交换代数，凸和离散几何形状，生成函数以及通常在优化标准课程之外的其他工具的方法支持的方法支持的方法。离散优化理论中的代数和几何思想提供了几种研究技术，在离散优化的实践者中尚不广为人知，最大程度地减少学习这些方法的先决条件，并提供了从线性离散优化到非线性离散优化的过渡。受众：这本书可以用作高级本科生的教科书，也可以用作数学，计算机科学或运营研究的初学者，也可以用作数学家，工程师和科学家的教程，他们希望更深入地研究如何以及为什么如何深入了解如何以及为什么算法确实或不起作用。内容：第一部分：已建立的离散优化工具；第1章：线性和凸优化的工具；第2章：数字和整数优化的工具；第二部分：GRAVER基础方法；第3章：Graver Bases；第4章：块结构整数程序的GRAVER基础；第三部分：生成功能方法；第5章：生成功能简介；第6章：多面体指标函数的分解；第7章：Barvinok的短期理性生成功能；第8章：通过求和全局混合组合多项式优化；第9章：通过整数投影进行多标准整数线性优化；第四部分：GRBNER基础方法；第10章：多项式计算；第11章：整数编程中的Grbner基础；第五部分：nullstellensatz和potitivstellensatz放松；第12章：离散优化中的零史塔兹；第13章：多项式和全局优化的阳性；第14章：结语。
This book presents recent advances in the mathematical theory of discrete optimization, particularly those supported by methods from algebraic geometry, commutative algebra, convex and discrete geometry, generating functions, and other tools normally considered outside the standard curriculum in optimization. Algebraic and Geometric Ideas in the Theory of Discrete Optimization offers several research technologies not yet well known among practitioners of discrete optimization, minimizes prerequisites for learning these methods, and provides a transition from linear discrete optimization to nonlinear discrete optimization. Audience: This book can be used as a textbook for advanced undergraduates or beginning graduate students in mathematics, computer science, or operations research or as a tutorial for mathematicians, engineers, and scientists engaged in computation who wish to delve more deeply into how and why algorithms do or do not work. Contents: Part I: Established Tools of Discrete Optimization; Chapter 1: Tools from Linear and Convex Optimization; Chapter 2: Tools from the Geometry of Numbers and Integer Optimization; Part II: Graver Basis Methods; Chapter 3: Graver Bases; Chapter 4: Graver Bases for Block-Structured Integer Programs; Part III: Generating Function Methods; Chapter 5: Introduction to Generating Functions; Chapter 6: Decompositions of Indicator Functions of Polyhedral; Chapter 7: Barvinok s Short Rational Generating Functions; Chapter 8: Global Mixed-Integer Polynomial Optimization via Summation; Chapter 9: Multicriteria Integer Linear Optimization via Integer Projection; Part IV: Grbner Basis Methods; Chapter 10: Computations with Polynomials; Chapter 11: Grbner Bases in Integer Programming; Part V: Nullstellensatz and Positivstellensatz Relaxations; Chapter 12: The Nullstellensatz in Discrete Optimization; Chapter 13: Positivity of Polynomials and Global Optimization; Chapter 14: Epilogue.